THE RULES FOR DIFFERENTIATION 89 



When we differentiate the logarithm of any function of x, we 

 get a fraction the denominator of which is that function of x and 

 the numerator is the differential coefficient of the denominator. 



For if t/ = log e /(,r), 



then y = loggZ where z=f(x) 



dy 1 dz 



But -* 



ax dz dx 



/(*) 



(6) To differentiate the sum of a certain number of functions 

 of x. 



For iiy = u+v+w + . . . where u, v, w . . . are functions of x, 

 then y + By = (u + Bu) + (v + Bv) + (w + Bw) + . . . 



and By = Bu + Bv + Brv + . . . 



By Bu Bv Bw 



& __ I ___ I ___ l_ 



Bx 8x Bx Bx 

 Making Bx infinitely small, 



dy du dv dw 



y = __ | ___ i ___ _i_ 



dx dx dx dx 



Hence the differential coefficient of the sum of a certain number 

 of functions of x is the sum of the differential coefficients of each 

 function. 



Thus if y = a + bVx + 7= + dx 3 



Vx 



Then y = a + bx* + ex + da? 



dy 1. J- 1 1 -I- 1 



-r = + d> x ~ T X 

 dx 2 2 



1, -i l -| 

 = -bx -ex + Soar 



^ - 



*T~ OCt^C 



